PROCEEDINGS
OF THE
AMERICAN MATHEMATICAL
Volume
89, Number
2, October
SOCIETY
1983
SUMSOF THREE INTEGER SQUARES
IN COMPLEX QUADRATICFIELDS
DENNIS R. ESTES AND J. S. HSIA1
Abstract.
We classify all complex quadratic number fields that have all their
algebraic integers expressible as a sum of three integer squares. These fields
are F = Q(V—D), D a positive square-free integer congruent to 3 (mod 8) and
such that D does not admit a positive proper factorization
D = di<¿2 that
satisfies simultaneously:
d\ = 5,7(mod8)
and ((¿2/di) = 1.
We consider the following problem: Which complex quadratic number fields F =
Q(\/—D), D square-free, have all their algebraic integers expressible as a sum of three
integer squares?
The fact that all the algebraic integers in F must be representable as a sum
of 3 integer squares forces some necessary conditions on D. Firstly, —1 must be
so expressible implies that —1 is a sum of two squares in F. This is because the
minimal number ("Stufe") of representing —1 as a sum of squares in any formally
nonreal field is well known to be always a 2-power. On the other hand, those
number fields with Stufe < 2 are known, already to Hasse in [2], to be classified by
the property that all the local degrees are even at all the primes lying over 2Z. In the
present situation this means D^l
(mod8). Secondly, since every sum of squares
is congruent to a perfect square mod 2, we see that if 2Z were ramified in F then
there are algebraic integers which are incongruent to a square mod 2. Therefore,
the discriminant of F must be an odd integer. These conditions then force 2Z to
be inert in F, i.e., D = 3 (mod 8), which we shall assume henceforth.
Presently, the following partial results are known. Every integer in F may be
expressible as a sum of 3 integer squares when any of the following conditions is
fulfilled:
(i) the class number hp of F is odd [3, p. 532];
(ii) hF = 2 X (odd integer) [1];
(iii) D = <7i<72<73
where g¿ are primes congruent to 3 (mod 4) and (QiÇj/Çfc)= —1
for {i,j,k} = {1,2,3} [1];
(iv) X2 + 2Y2 = D is solvable in Z [6, p. 7].
In terms of factorizations of D, the classical theorems of Gauss and Rédei-Reichardt
imply respectively that (i) is equivalent to D being a prime and (ii) to D = pq where
p,q are primes and (p/q) = —1. As for (iv) it is well known that the only odd
square-free positive integers (and hence D) which can be represented by the binary
form X2 + 2Y2 are those whose prime factors are either congruent to 1 or to 3
(mod8).
Received by the editors October 15, 1982.
1980 Mathematics Subject Classification. Primary 10C02, 10C05.
Key words and phrases. Exceptional
integer, genus, x-invariant,
Artin symbol.
Research partially supported by NSF Grant MCS 80-02985 and MCS 82-02065.
©1983 American Mathematical
Society
0002-9939/83 $1.00 + $.25 per page
211
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212
D. R. ESTES AND J. S. HSIA
In fact, it was shown in [1] that in the first three cases above every algebraic
integer in F is representable by every quadratic form in the genus of X2 +Y2 +X2.
On the negative side, however, one knows not every integer is a sum of 3 integer
squares for certain fields whose ideal class groups are either the Klein 4-group or
the cyclic group of order four. See [1].
We shall give a complete answer to our problem, which is the following
Theorem.
Every algebraic integer in F = Q(\/—D), D a positive square-free
integer, can be expressed as a sum of three integer squares when and only when D =
3 (mod8) andD does not admit a positive proper factorization D = djc^ (i.e., d» > I)
which satisfies the conditions: (1) di = 5,7 (mod 8) and (2) (d2/di) = 1.
We may extract the following immediate corollary which is more explicit when
the Sylow 2-subgroup Cp(2) of the ideal class group Cp is cyclic.
Corollary.
Let F be as in the theorem with D = 3 (mod8). Suppose Cp(2)
is cyclic. Then, every integer in F is a sum of three integer squares if and only if
D satisfies any of the three conditions: (a) D is a prime; (b) D = pq, p, q primes,
(p/q) = —1; and (c) D = pq, p, q primes, (p/q) = I, and p = 1 (mod8), q = 3 (mod8).
We now see that all the previously known results are recaptured. Results (i) and
(ii) are covered by the corollary. As for (iv), since the only prime factors of D are
those congruent to either 1 or 3 (mod 8), no such proper factorizations as required
by the theorem are possible. To see result (iii), suppose ft — 92 — 7 (mod 8). Then
di can be either <7i,<72,<7i93,
or q2q3 so that violating condition (2) of the Theorem
forces result (iii).
Next, we observe that since —1 is a sum of 2 squares in F the quadratic space
associated to the form Is = X2 +Y2 + Z2 must be isotropic, and therefore, the
theoretical machinery derived in [1] may be applied. In particular, we can extract
the following special cases of the theorems in [1, §§1, 2] for the genus of Is over
F = Q(v/=^).
Fact I. A nonzero integer c of F is an exceptional integer for the genus of I¡
(i.e., c is representable by some, but not by every form in the genus) if and only if.
(1.1) —c is a nonsquare in F;
(1.2) —c is a square in F2;
(1.3)F(y/^c)/F
is unramifitd;
(1.4) writing (c) = (p"x- • -p"')2, a3 > 0, then at each j the Artin symbol
fF(y/=-c)/F\
L
Fact II. A nonzero integere of F is an exceptional integer for I3 (i.e., some form
in the genus of I3, but not I3 itself, represents c) if and only if.
(n.l) —c is a nonsquare in F;
(n.2) every prime dividing (2c) splits in F(y/^c);
(n.3) F(,/:ic)/F
(n.4) ifxih)
is unramified;
= ACF then A is not a norm from F(v/—c)-
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SUMS OF THREE INTEGER SQUARES
213
Here A denotes the ideal class of A. For the definition of the x-invariant, see [4,
p. 329]. For our purpose here, it suffices to know that x(h) = ACF where A is the
coefficient ideal of any isotropic vector.
Remark.
Fact LI solves the finer problem of whether a particular algebraic
integer c in F is or is not a sum of 3 integer squares. This can be exploited to
pinpoint explicitly an integer which is not a sum of 3 integer squares. For example,
take the cyclic Sylow 2-subgroup case where D = pq, p,q primes, (p/q) — 1, p =
5(mod8) and q= 1 (mod8). One checks that c = q is an exception for I3.
Proof of Theorem.
View the form I3 as a free ternary lattice with orthonormal basis {ei,e2,e3}. If v is an isotropic vector with v = aiei +a2e2 + 0^3, then
the coefficient ideal A of v in I3 is given by
(ofx) n (af1) n (a^1) = (ai) + (a2) + (a3).
See [5, p. 212]. Let R denote the ring of algebraic integers in F. Then, J2i=i aiR ™
generated by any two of the a^. To see this, we localize and assume R is a discrete
valuation ring. Put aR = J2i=i ai^ an(l ai = a^i- We may assume that not all a¿
are zero so that one of í¿ is a unit. But, J212 = 0- So, at least two of the í¿ must
be units which proves the claim.
Suppose r,s,t,w EX satisfy the equation Dr2 = s2 +t2 + w2. Then,
(s2 +12)2 + (sw + trV^-D)2 + (tw - sr^f^-D)2 = 0
gives rise to an isotropic vector.
Hence, x(^)
= ACF where A = (s2 + t2)R +
(sw + trVIID)R.
Consider the binary quadratic form f(X,Y) = -2X2 + 2XY + ((D - 1)/2)Y2.
Since D = 3 (mod 8), this form is primitive. Hence, it represents an odd prime p
prime to any fixed integer of our choice (see e.g. Dickson's History, Vol. I, p. 417).
We take p prime to D. As 2p = 2f(x, y) = -(2z - y)2 + Dy2 = -1 + 3 (mod 8), p = 1
(mod4). Set 2p = s2 + t2, 2x-y
and x(^3) = AC2F, where
= w and y = r. Then, Dr2 = s2 + t2 + w2 = 2p + w2
Now,
1 *•> / 7-. (sw + tr\f^D\\(
NF/Çi(A)
= lPR + l\R\lpR„ + fsw-tr\f^D\r\
l-^-
\R\
( _
_ (w2 + t2\^
fsw + tr\/-D\S\
= pipR + swR + (-^—
)R + l-^\R\ = pR
since pR + swR + ((w2 + t2)/2)R = R. We conclude, therefore, that A is a prime
ideal of norm pi.
Suppose c is an exceptional integer for Í3, then it is surely exceptional for the
genus of 73. Set (c) = B2 as in (1.4). Then B contains an ambiguous ideal E. Write
B = bE and E2 = (di), where di\D and c = udib2 for some unit in R. We may
assume clearly that D > 3. Hence, u = +1. Replacing di by —di, if needed, we
may suppose that c = dio2. By (1.1), di # —1, D and di = 3(mod4) by (1.2). Since
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D. R. ESTES AND J. S. HSIA
214
F(i/=c) = F(y/—di) = F(y/d2), did2 = D, condition (1.4) implies that
(f(s/t2)/f\
V E
f d2 y
) [\di\J l-
Next, condition (LT.4)
(Fy/=c)IF\ ' f-dA f-2\
{ A J V V ) \\di\J
Conversely, assume there is a factorization
D = did2 satisfying di = 3 (mod4),
di jt —1, D, (d2/|di|) = 1 and (—2/|di|) = —1. Set (di) = E2 and choose a prime
ideal P in Ê and write P = bE and c = di&2. Then (c) = P2. Clearly, —c is a
nonsquare in F. Since di = 3 (mod 4), —c is a square in F2. Condition
consequence of the equalities F(y/~=c) = F(y/—di) = F(\fd2¡. Hence,
(LI.3) is a
So, we see (LI.2) is satisfied. Finally, we note that (—2/|di|) = —1 implies condition
(n.4) is also met. This proves c is an exceptional integer for I3.
Recapturing, we have proved that ^3 has an exceptional integer if and only if
D admits a factorization D = did2 where (i) di = 3 (mod4), (ii) di # —1, D, (iii)
(d2/|di |) = 1 and (iv) (—2/|di|) = —1. If we require the factorization of D to be
positive proper then we get exactly the conditions as stated in our Theorem.
References
1. D. R. Estes and J. S. Hsia, Exceptional integers 0} some ternary quadratic forms, Adv. in Math.
45 (1982), 310-318.
2. H. Hasse, Darstellbarkeit
von Zahlen durch quadratische Formen in einem beliebigen algebraischen
Zahlkorper,J. Reine Angew. Math. 153 (1924), 113-130.
3. J. S. Hsia,
Representations
by integral quadratic forms over algebraic number fields, Queen's
Papers in Pure and Appl. Math. 46 (1977), 528-537.
4. _,
On the classification ofunimodular quadratic forms, J. Number Theory 12 (1980), 327-333.
5. O. T. O'Meara, Introduction to quadratic forms, Grundlehren der Math. Wiss., Springer-Verlag,
New York, 1963.
6. P. Revoy, Sur les sommes de carrés dans un anneau, Ann. Sei. Univ. Besançon
Math.
(3) 11
(1979), 3-8.
Department
of
Angeles, California
Department
Mathematics,
90089-1113
of Mathematics,
University
Ohio State
of
University,
Southern
Columbus,
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California,
Ohio 43210
Los